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187

Level

Mathematics89

AdditionBasic arithmeticBeginner

a + b = c

Combining two numbers.

SubtractionBasic arithmeticBeginner

a - b = c

Finding the unknown addend.

MultiplicationBasic arithmeticBeginner

a \times b = c

Repeated addition.

DivisionBasic arithmeticBeginner

a \div b = c \quad (b \neq 0)

Finding the unknown factor.

Commutative property of additionPropertiesBeginner

a + b = b + a

Swapping addends does not change the sum.

Associative property of additionPropertiesBeginner

(a + b) + c = a + (b + c)

Grouping does not change the sum.

Distributive propertyPropertiesBeginner

a \times (b + c) = a \times b + a \times c

Multiplication distributes over addition.

Arithmetic meanStatisticsBeginner

\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}

Sum divided by count.

PercentageRatios & proportionsBeginner

p = \frac{\text{部分}}{\text{整体}} \times 100\%

A part per hundred.

Speed–distance–timeWord problemsBeginner

v = \frac{s}{t}

Distance equals speed times time.

Perimeter of a rectangleGeometryBeginner

C = 2(a + b)

Where $a$, $b$ are length and width.

Area of a rectangleGeometryBeginner

S = ab

Length times width.

Area of a triangleGeometryBeginner

S = \frac{1}{2}ah

Half of base times height.

Area of a parallelogramGeometryBeginner

S = ah

Base times height.

Area of a trapezoidGeometryBeginner

S = \frac{1}{2}(a + b)h

Average of parallel sides times height.

Circumference of a circleGeometryBeginner

C = 2\pi r

Where $r$ is the radius.

Area of a circleGeometryBeginner

S = \pi r^2

Pi times radius squared.

Volume of a cuboidGeometryBeginner

V = abc

Length times width times height.

Volume of a cylinderGeometryBeginner

V = \pi r^2 h

Base area times height.

Volume of a coneGeometryBeginner

V = \frac{1}{3}\pi r^2 h

One third of the cylinder volume.

Volume of a sphereGeometryBeginner

V = \frac{4}{3}\pi r^3

Where $r$ is the radius.

Surface area of a sphereGeometryBeginner

S = 4\pi r^2

Surface area of a sphere.

Lateral area of a cylinderGeometryBeginner

S = 2\pi r h

Circumference times height.

Pythagorean theoremGeometryMiddle

a^2 + b^2 = c^2

In a right triangle, the sum of squares of legs equals the square of the hypotenuse.

Linear equationAlgebraMiddle

ax + b = 0 \quad (a \neq 0)

The solution is $x = -b/a$.

ProportionRatios & proportionsMiddle

\frac{a}{b} = \frac{c}{d}

Cross-multiplication gives $ad = bc$.

Pythagorean identityTrigonometryMiddle

\sin^2\theta + \cos^2\theta = 1

The fundamental trigonometric identity.

Tangent identityTrigonometryMiddle

\tan\theta = \frac{\sin\theta}{\cos\theta}

Tangent equals sine over cosine.

Triangle area (sine form)TrigonometryMiddle

S = \frac{1}{2}ab\sin C

Area from two sides and the included angle.

Law of sinesTrigonometryMiddle

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Where $R$ is the circumradius.

Law of cosinesTrigonometryMiddle

a^2 = b^2 + c^2 - 2bc\cos A

A generalisation of Pythagoras.

Sine addition formulaTrigonometryHigh

\sin(A+B) = \sin A\cos B + \cos A\sin B

Sine of a sum.

Cosine addition formulaTrigonometryHigh

\cos(A+B) = \cos A\cos B - \sin A\sin B

Cosine of a sum.

Sine double angleTrigonometryHigh

\sin 2\theta = 2\sin\theta\cos\theta

Special case of addition with $A=B=\theta$.

Cosine double angleTrigonometryHigh

\cos 2\theta = \cos^2\theta - \sin^2\theta

Also $2\cos^2\theta - 1$ or $1 - 2\sin^2\theta$.

Quadratic formulaAlgebraHigh

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The discriminant $\Delta = b^2 - 4ac$ decides the number of roots.

Vieta’s formulasAlgebraHigh

x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

The relation between roots and coefficients.

Binomial theoremAlgebraHigh

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

The general term is $\binom{n}{k}a^{n-k}b^k$.

AM–GM inequalityInequalitiesHigh

\frac{a+b}{2} \geq \sqrt{ab} \quad (a, b > 0)

Equality holds exactly when $a = b$.

Triangle inequalityInequalitiesHigh

|a + b| \leq |a| + |b|

Generalises to $n$ terms.

Distance between two pointsAnalytic geometryHigh

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Distance in the Cartesian plane.

Equation of a circleAnalytic geometryHigh

(x - a)^2 + (y - b)^2 = r^2

Centre $(a, b)$, radius $r$.

Slope of a lineAnalytic geometryHigh

k = \frac{y_2 - y_1}{x_2 - x_1}

Slope from two points.

Point-slope formAnalytic geometryHigh

y - y_0 = k(x - x_0)

Line through a point with given slope.

Exponent product ruleExponents & logsHigh

a^m \cdot a^n = a^{m+n}

Same base: add exponents.

Power of a powerExponents & logsHigh

(a^m)^n = a^{mn}

Power of a power: multiply exponents.

Zero exponentExponents & logsHigh

a^0 = 1 \quad (a \neq 0)

Any non-zero number to the power of zero is 1.

Negative exponentExponents & logsHigh

a^{-n} = \frac{1}{a^n}

A negative exponent means reciprocal.

Logarithm definitionExponents & logsHigh

a^y = x \;\Leftrightarrow\; y = \log_a x \quad (a > 0,\, a \neq 1)

Logarithm is the inverse of exponentiation.

Logarithm product ruleExponents & logsHigh

\log_a(xy) = \log_a x + \log_a y

Log of a product equals sum of logs.

Logarithm quotient ruleExponents & logsHigh

\log_a \frac{x}{y} = \log_a x - \log_a y

Log of a quotient equals difference of logs.

Logarithm power ruleExponents & logsHigh

\log_a x^n = n \log_a x

Log of a power: exponent comes out front.

Change of baseExponents & logsHigh

\log_b a = \frac{\log_c a}{\log_c b}

Can switch to any base $c$.

PermutationsCombinatoricsHigh

A_n^r = \frac{n!}{(n-r)!}

Arrangements of $r$ from $n$.

CombinationsCombinatoricsHigh

C_n^r = \frac{n!}{r!(n-r)!}

Selections without order.

Arithmetic sequenceSequencesHigh

a_n = a_1 + (n-1)d

Where $d$ is the common difference.

Geometric sequenceSequencesHigh

a_n = a_1 r^{n-1}

Where $r$ is the common ratio.

Arithmetic series sumSequencesHigh

S_n = \frac{n(a_1 + a_n)}{2}

Average of first and last times count.

Geometric series sumSequencesHigh

S_n = \frac{a_1(1 - r^n)}{1 - r} \quad (r \neq 1)

Valid for $r \neq 1$.

Infinite geometric seriesSequencesHigh

S = \frac{a_1}{1 - r} \quad (|r| < 1)

Converges when $|r| < 1$.

Complex numberComplex numbersHigh

z = a + bi

Real part $a$, imaginary part $b$, $i^2 = -1$.

Modulus of a complex numberComplex numbersHigh

|z| = \sqrt{a^2 + b^2}

Distance from origin in the complex plane.

Definition of the derivativeCalculusHigh

f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}

Geometrically the tangent slope.

Power ruleCalculusHigh

(x^n)' = n x^{n-1}

The most used differentiation rule.

Product ruleCalculusHigh

(uv)' = u'v + uv'

For differentiating a product.

Quotient ruleCalculusHigh

\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

For differentiating a quotient.

Chain ruleCalculusHigh

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

The core of differentiating composites.

Common derivativesCalculusHigh

(\sin x)' = \cos x,\; (\cos x)' = -\sin x,\; (e^x)' = e^x,\; (\ln x)' = \frac{1}{x}

Four frequently used derivatives.

Newton–Leibniz formulaCalculusHigh

\int_a^b f(x)\,dx = F(b) - F(a)

Where $F$ is an antiderivative of $f$.

Integration by partsCalculusHigh

\int u\,dv = uv - \int v\,du

For integrals of products.

Taylor expansionCalculusUG Basic

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n

With $a=0$ it is a Maclaurin series.

L’Hôpital’s ruleCalculusUG Basic

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

For $0/0$ or $\infty/\infty$ limits.

Euler’s formulaComplex numbersUG Advanced

e^{i\theta} = \cos\theta + i\sin\theta

The bridge between exponentials and trigonometry.

De Moivre’s theoremComplex numbersUG Advanced

(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)

A corollary of Euler’s formula.

Gaussian integralProbabilityUG Advanced

\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}

Key to normalising the normal distribution.

Bayes’ theoremProbabilityUG Advanced

P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}

Inferring causes from evidence.

Cauchy–Schwarz inequalityInequalitiesUG Advanced

\left(\sum_{i=1}^n a_i b_i\right)^2 \leq \left(\sum_{i=1}^n a_i^2\right)\left(\sum_{i=1}^n b_i^2\right)

A fundamental inequality in inner-product spaces.

Jensen’s inequalityInequalitiesUG Advanced

f\!\left(\sum \lambda_i x_i\right) \leq \sum \lambda_i f(x_i)

For convex $f$ with $\sum\lambda_i=1$.

Mean value theoremCalculusUG Advanced

f(b) - f(a) = f'(\xi)(b - a), \quad \xi \in (a, b)

Links function values to the derivative.

Rolle’s theoremCalculusUG Advanced

f(a) = f(b) \Rightarrow \exists\,\xi \in (a,b),\; f'(\xi) = 0

A special case of the MVT.

Determinant of a 2×2 matrixLinear algebraUG Advanced

\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

The simplest determinant.

Cramer’s ruleLinear algebraUG Advanced

x_i = \frac{D_i}{D}

Solving linear systems with determinants.

Eigenvalue equationLinear algebraUG Advanced

A\mathbf{v} = \lambda\mathbf{v}

A vector whose direction is preserved by $A$.

Fourier seriesAnalysisUG Advanced

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}\left(a_n \cos nx + b_n \sin nx\right)

Decomposing a periodic function into sinusoids.

Laplace transformAnalysisUG Advanced

\mathcal{L}\{f(t)\} = \int_0^{\infty} f(t)\,e^{-st}\,dt

Turns ODEs into algebraic equations.

Stokes’ theoremAnalysisUG Advanced

\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}

Relates line and surface integrals.

GradientAnalysisUG Advanced

\nabla f = \left(\frac{\partial f}{\partial x},\, \frac{\partial f}{\partial y},\, \frac{\partial f}{\partial z}\right)

Points in the direction of steepest ascent.

DivergenceAnalysisUG Advanced

\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}

Source or sink strength of a vector field.

Riemann sum definitionAnalysisUG Advanced

\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i)\,\Delta x

The limit definition of the integral.

Physics45

SpeedMechanicsBeginner

v = \frac{s}{t}

Distance per unit time.

DensityMechanicsBeginner

\rho = \frac{m}{V}

Mass per unit volume.

WeightMechanicsBeginner

G = mg

Where $g \approx 9.8\;\text{m/s}^2$.

PressureMechanicsBeginner

p = \frac{F}{S}

Force per unit area.

Hydrostatic pressureMechanicsBeginner

p = \rho g h

Where $h$ is the depth.

BuoyancyMechanicsBeginner

F_{\text{浮}} = \rho g V_{\text{排}}

Weight of displaced fluid.

WorkMechanicsBeginner

W = Fs

Force times displacement.

PowerMechanicsBeginner

P = \frac{W}{t}

Work per unit time.

Lever principleMechanicsBeginner

F_1 l_1 = F_2 l_2

Effort times arm equals load times arm.

HeatThermodynamicsBeginner

Q = cm\Delta t

Where $c$ is specific heat capacity.

Ohm’s lawElectromagnetismBeginner

I = \frac{U}{R}

Current is proportional to voltage, inverse to resistance.

Series resistanceElectromagnetismBeginner

R = R_1 + R_2 + \cdots + R_n

Sum of individual resistances.

Parallel resistanceElectromagnetismBeginner

\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}

Reciprocal of total equals sum of reciprocals.

Electric powerElectromagnetismBeginner

P = UI = I^2 R = \frac{U^2}{R}

Three equivalent forms.

Joule’s lawElectromagnetismBeginner

Q = I^2 R t

Heat from electric current.

Newton’s second lawMechanicsHigh

\vec{F} = m\vec{a}

Net force determines acceleration.

Uniformly accelerated motionMechanicsHigh

v = v_0 + at,\;\; s = v_0 t + \tfrac12 at^2,\;\; v^2 - v_0^2 = 2as

The three kinematic equations.

Work–energy theoremMechanicsHigh

W_{\text{合}} = \tfrac12 mv_2^2 - \tfrac12 mv_1^2

Net work equals the change in kinetic energy.

Conservation of momentumMechanicsHigh

m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'

For a system with zero net external force.

Law of universal gravitationMechanicsHigh

F = G\frac{m_1 m_2}{r^2}

$G = 6.67\times10^{-11}\;\text{N·m}^2/\text{kg}^2$.

Centripetal forceMechanicsHigh

F = m\frac{v^2}{r}

Net force toward the centre of the circle.

Centripetal accelerationMechanicsHigh

a = \frac{v^2}{r} = \omega^2 r

Directed toward the centre.

Angular velocityMechanicsHigh

\omega = \frac{2\pi}{T} = 2\pi f

Where $T$ is the period and $f$ the frequency.

Simple pendulum periodMechanicsHigh

T = 2\pi\sqrt{\frac{l}{g}}

Valid for small-angle oscillations.

Simple harmonic motionVibration & wavesHigh

x = A\cos(\omega t + \varphi),\;\; \omega = \sqrt{\frac{k}{m}}

Displacement varies as a cosine.

Wave speedVibration & wavesHigh

v = \lambda f

Speed equals wavelength times frequency.

Doppler effectVibration & wavesHigh

f' = f\,\frac{v \pm v_o}{v \mp v_s}

Frequency shift from relative motion.

Ideal gas lawThermodynamicsHigh

pV = nRT

$R = 8.314\;\text{J/(mol·K)}$.

First law of thermodynamicsThermodynamicsHigh

\Delta U = W + Q

Internal energy change equals work plus heat.

Coulomb’s lawElectromagnetismHigh

F = k\frac{q_1 q_2}{r^2}

$k = 9\times10^9\;\text{N·m}^2/\text{C}^2$.

Lorentz forceElectromagnetismHigh

F = qvB\sin\theta

Direction from the left-hand rule.

Faraday’s lawElectromagnetismHigh

\varepsilon = -N\frac{d\Phi}{dt}

The sign indicates opposition to the change in flux.

Magnetic fluxElectromagnetismHigh

\Phi = BS\cos\theta

Where $\theta$ is the angle to the normal.

CapacitanceElectromagnetismHigh

C = \frac{Q}{U}

In farads (F).

Energy stored in a capacitorElectromagnetismHigh

E = \frac{1}{2}CU^2

Energy stored in the electric field.

Resistance & resistivityElectromagnetismHigh

R = \rho\frac{l}{S}

Where $\rho$ is resistivity.

Snell’s lawOpticsHigh

n_1 \sin\theta_1 = n_2 \sin\theta_2

Light passing between media.

Thin lens equationOpticsHigh

\frac{1}{u} + \frac{1}{v} = \frac{1}{f}

Relates object, image and focal distances.

Photoelectric effectModern physicsHigh

E_k = h\nu - W

Photon energy minus work function.

de Broglie wavelengthModern physicsHigh

\lambda = \frac{h}{p}

Wave nature of particles.

Mass–energy equivalenceModern physicsUG Advanced

E = mc^2

$c = 3\times10^8\;\text{m/s}$.

Uncertainty principleModern physicsUG Advanced

\Delta x\,\Delta p \geq \frac{\hbar}{2}

Position and momentum cannot both be exact.

Schrödinger equationModern physicsUG Advanced

i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi

The central equation of quantum mechanics.

Planck relationModern physicsUG Advanced

E = h\nu

Photon energy is proportional to frequency.

Stefan–Boltzmann lawThermodynamicsUG Advanced

j = \sigma T^4

Black-body radiance scales as $T^4$.

Chemistry20

Amount of substanceAmount of substanceMiddle

n = \frac{m}{M} = \frac{N}{N_A} = \frac{V}{V_m}

$N_A = 6.02\times10^{23}\,\text{mol}^{-1}$.

Molar concentrationSolutionsMiddle

c = \frac{n}{V}

In mol/L.

Dilution lawSolutionsMiddle

c_1 V_1 = c_2 V_2

The amount of solute is unchanged by dilution.

Mass fractionSolutionsMiddle

w = \frac{m_{\text{质}}}{m_{\text{液}}} \times 100\%

Percentage of solute mass in solution.

Definition of pHAcids & basesMiddle

\mathrm{pH} = -\lg[\mathrm{H^+}]

Below 7 is acidic, above 7 basic.

Equilibrium constantEquilibriumHigh

K = \frac{[\mathrm{C}]^c[\mathrm{D}]^d}{[\mathrm{A}]^a[\mathrm{B}]^b}

Depends only on temperature.

Rate lawKineticsHigh

v = k[A]^m[B]^n

Where $k$ is the rate constant.

Arrhenius equationKineticsHigh

k = A\,e^{-E_a/(RT)}

A smaller activation energy gives a larger $k$.

Integrated first-order lawKineticsHigh

[A] = [A]_0\,e^{-kt},\;\; t_{1/2} = \frac{\ln 2}{k}

The half-life is independent of concentration.

Hess’s lawThermochemistryHigh

\Delta H = \sum \Delta H_i

Enthalpy depends only on the endpoints.

Nernst equationElectrochemistryHigh

E = E^\circ - \frac{RT}{nF}\ln Q

Relates electrode potential to concentration.

Faraday’s law of electrolysisElectrochemistryHigh

m = \frac{MQ}{nF}

Where $Q$ is the charge passed.

Colligative propertiesSolutionsHigh

\Delta T_f = K_f\,b

Freezing-point depression is proportional to molality.

Buffer pH (Henderson–Hasselbalch)Acids & basesHigh

\mathrm{pH} = \mathrm{p}K_a + \lg\frac{[\mathrm{A^-}]}{[\mathrm{HA}]}

Buffers resist pH change.

Solubility productEquilibriumHigh

K_{sp} = [A^m]^n[B^n]^m

For sparingly soluble salts.

Gibbs free energyThermodynamicsUG Basic

\Delta G = \Delta H - T\Delta S

Spontaneous when $\Delta G < 0$.

Beer–Lambert lawAnalytical chemistryUG Basic

A = \varepsilon b c

Absorbance proportional to concentration.

van ’t Hoff equationThermodynamicsUG Advanced

\ln\frac{K_2}{K_1} = \frac{\Delta H^\circ}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)

Temperature dependence of $K$.

Clausius–Clapeyron equationThermodynamicsUG Advanced

\ln\frac{P_2}{P_1} = -\frac{\Delta H_{\text{vap}}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)

Vapour pressure vs temperature.

Boltzmann distributionStatistical thermodynamicsUG Advanced

\frac{n_i}{n_0} = e^{-E_i/(kT)}

Population ratio of energy levels.

Biology12

PhotosynthesisMetabolismMiddle

\mathrm{6CO_2 + 6H_2O} \xrightarrow[\text{叶绿体}]{\text{光}} \mathrm{C_6H_{12}O_6 + 6O_2}

Light energy into chemical energy.

Aerobic respirationMetabolismMiddle

\mathrm{C_6H_{12}O_6 + 6O_2} \to \mathrm{6CO_2 + 6H_2O} + \text{能量}

Organic matter fully oxidised to release energy.

DNA base pairingGeneticsMiddle

\mathrm{A\text{-}T},\;\; \mathrm{G\text{-}C}

Adenine pairs with thymine, guanine with cytosine.

Hardy–WeinbergGeneticsHigh

p^2 + 2pq + q^2 = 1

Allele frequencies stay constant in an ideal population.

Exponential growthEcologyHigh

N_t = N_0\,e^{rt}

Population growth with abundant resources.

Logistic growthEcologyHigh

\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)

Where $K$ is the carrying capacity.

Trophic efficiencyEcologyHigh

\eta = \frac{E_{n+1}}{E_n} \times 100\%

Typically 10–20%.

Michaelis–MentenEnzyme kineticsHigh

v = \frac{V_{\max}[S]}{K_m + [S]}

$K_m$ reflects substrate affinity.

Central dogmaMolecular biologyUG Basic

\mathrm{DNA} \xrightarrow{\text{转录}} \mathrm{RNA} \xrightarrow{\text{翻译}} \mathrm{蛋白质}

The flow of genetic information.

Shannon diversity indexEcologyUG Basic

H' = -\sum_{i=1}^n p_i \ln p_i

Where $p_i$ is the proportion of species $i$.

Chi-square testBiostatisticsUG Basic

\chi^2 = \sum\frac{(O_i - E_i)^2}{E_i}

Compares observed with expected values.

Fick’s first lawPhysiologyUG Advanced

J = -D\frac{dC}{dx}

Describes diffusion flux.

Computer Science21

Binary to decimalBasicsBeginner

N_{10} = \sum_{i=0}^{n} d_i \times 2^i

Each bit times a power of two.

Big-O orderingAlgorithm analysisMiddle

O(1) < O(\log n) < O(n) < O(n\log n) < O(n^2) < O(2^n)

From fastest to slowest growth.

Binary search midpointAlgorithmsMiddle

\text{mid} = \left\lfloor\frac{\text{low} + \text{high}}{2}\right\rfloor

Core of searching a sorted array.

Hash table load factorData structuresMiddle

\alpha = \frac{n}{m}

Elements per bucket.

Cache hit ratioSystemsMiddle

h = \frac{N_{\text{hit}}}{N_{\text{hit}} + N_{\text{miss}}}

Fraction of accesses that are hits.

Master theoremAlgorithm analysisHigh

T(n) = aT(n/b) + f(n)

For analysing divide-and-conquer recurrences.

Naive BayesMachine learningHigh

P(y \mid x) \propto P(y)\prod_i P(x_i \mid y)

Assumes conditional independence of features.

Cross-entropy lossMachine learningHigh

L = -\frac{1}{N}\sum_i\left[y_i \ln\hat{y}_i + (1-y_i)\ln(1-\hat{y}_i)\right]

The usual loss for classification.

Gradient descentMachine learningHigh

\theta \leftarrow \theta - \eta\,\nabla L(\theta)

Here $\eta$ is the learning rate.

Sigmoid functionMachine learningHigh

\sigma(x) = \frac{1}{1 + e^{-x}}

Maps real numbers to $(0,1)$.

Linear regressionMachine learningHigh

\hat{y} = \mathbf{w}^\top \mathbf{x} + b

The simplest regression model.

Mean squared errorMachine learningHigh

\mathrm{MSE} = \frac{1}{n}\sum_{i=1}^n (y_i - \hat{y}_i)^2

Common loss for regression.

ReLU activationDeep learningHigh

\mathrm{ReLU}(x) = \max(0, x)

The most common activation in deep nets.

Scaled dot-product attentionDeep learningUG Basic

\mathrm{Attention}(Q,K,V) = \mathrm{softmax}\!\left(\frac{QK^\top}{\sqrt{d_k}}\right)V

The core operation of a Transformer.

SoftmaxDeep learningUG Basic

\sigma(z)_i = \frac{e^{z_i}}{\sum_j e^{z_j}}

Maps a real vector to a probability distribution.

Batch normalizationDeep learningUG Basic

\hat{x} = \frac{x - \mu_B}{\sqrt{\sigma_B^2 + \epsilon}}

Accelerates training convergence.

Amdahl’s lawPerformanceUG Basic

S = \frac{1}{(1-p) + \frac{p}{s}}

Speed-up is bounded by the serial part.

Bandwidth-delay productNetworkingUG Basic

\mathrm{BDP} = \mathrm{RTT} \times \text{带宽}

Caps a TCP connection’s throughput.

Shannon entropyInformation theoryUG Advanced

H = -\sum_{i=1}^n p_i \log_2 p_i

Expected information content.

Little’s lawQueueing theoryUG Advanced

L = \lambda W

Average number = arrival rate × sojourn time.

PageRankAlgorithmsUG Advanced

PR(i) = \frac{1-d}{N} + d\sum_{j \in B_i}\frac{PR(j)}{L(j)}

Where $d$ is the damping factor.

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